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Vol. 6, No. 1-2-3, 1995ISSN 1053-4792SMARANDACHE FUNCTION JOURNALS(n) is the smallest integer such that S(n)! is divisible by nS(n) is the smallest integer such that S(n)! is divisible by nS(n) is the smallest integer such that S(n)! is divisible by nNumber Theory Associationof theUNIVERSITY OF CRAIOVA

SFJ is periodically published in a 100-200 pages volume, and 8001000 copies.SFJ is a referred journal.SFJ is reviewed, indexed, cited by the following journals:"Zentralblatt Für Mathematik" (Germany), "Referativnyi Zhurnal"and "Matematika" (Academia Nauk, Russia), "Mathematical Reviews"(USA), "Computing Review" (USA), "Libertas Mathematica" (USA),"Octogon" (Romania), "Indian Science Abstracts" (India), "Ulrich'sInternational Periodicals Directory" (USA), "Gale Directory ofPublications and Broadcast Media" (USA), "Historia s"(USA),"TheMathematical Gazette" (U.K.), "Abstracts of Papers Presented tothe American Mathematical Society" (USA), "Personal ComputerWorld" (U.K.), "Mathematical Spectrum" (U.K.), "Bulletin of ficInformation (PA, USA), "Library of Congress Subject Headings"(USA).INDICATION TO AUTHORSAuthors of papers concerning any of Smarandache type functionsare encouraged to submit manuscripts to the Editors:Prof. C. Dumitrescu & Prof. V. SeleacuDepartment of MathematicsUniversity of Craiova, Romania.The submitted manuscripts may be in the format of remarks,conjectures, solved/unsolved or open new proposed problems, notes,articles, miscellaneous, etc.They must be original work andcamera ready [typewritten/computerized, format: 8.5 x 11 inches (.21,6 x 28 cm)].They are not returned, hence we advise theauthors to keep a copy.The title of the paper should be writing with capital letters.The author's name has to apply in the middle of the line, near thetitle. References should be mentioned in the text by a number insquare brackets and should be listed alphabetically.Currentaddress followed by e-mail address should apply at the end of thepaper, after the references.The paper should have at the beginning an up to a half-pageabstract, followed by the key words.All manuscripts are subject to anonymous review by two or moreindependent sources.2

On some numerical functionsMarcela Popescu, Paul Popescu and Vasile SeleacuDepartment of mathematics"University of Craiova13,A.I.Cuza st., Craiova, 1100, RomaniaIn this pa.per we prove that the following numerical functions:l!'(%)1. Fs: N- - N, Fs(x) Z S(pf), where Pi are the prime na.tural numbers whichi l (x)are not greater than x andis the number of them,2. 8 : N- - N, 8( x) Z S(pf) , where p, the prime na.tural numbers whichdivide x,3.e: N- -N, 8( x) Z S(pf) , where PiMe the prime na.tul&l numbers which MePi'l- %smaller than x and do not divide x,which involve the SmMandache function, does not verify the Lipschitz condition. Theseresults are useful to study the beha.viour of the numerical functions considered a.bove.",(z)Proposition 1 The function Fs: N- - N, Fs(x) Z S(pf), whe're Pi and (x)have1 1the signifience from a.bove, doel not verify the Lipschi:z condition.Proof. Let K 0 bea. given real number, x p be a prime natural number, which[.jKverify p 1} and y p - 1. It is easy to see that (p) (p - 1) 1, for everyprime natura.! number p, since the prime natura.! numbers which are not grea.ter than pMe the as those of (p - 1) in addition to p. We!Fs(x) - Fs(y)! Fs(p) - Fs(p - 1) [SC ) S( )"'''' S( (p l)) S(PP)] - [S( -l) S( -l) . S(? l))] 3

[S( ) - S(rl-1)] . [S{1' l)) - S(1' l)] T S{PP)·Bll S(pf) S(pf-l) for every i E 1, ,-(p - 1) , therefore we have!Fs( )- Fs(y)1 S(PP)·Beca.use S(PP) p2 , for every prime p, it follows:iF5{:) - Fs(Y)1 S(pP) p2 K K ·1 K (p - (p - 1)) K I: - yj.We proved tha. for every real K 0 there ex:is the na.tural numbers : P udy p - 1, choseD IS &hove, so hat lFs{:) - Fs(Y)1 K I: - YI, therefore Fs does no verify the Lipschitz condition.Remark 1 Another prooj, longer and more techn,ca can be made uling a rezuli whichassertl that the Smarandache junction S a.lso does not verify the Lipschitz conditiofL. Wehave choleft thi' proof becaUle it iI more simple and free of another multi.Proposition 2 The junctione : .V· - N,G( ) p,!s2: S(pf) , where P' are the primenatural number, which divide:, does not rJerify the Lipschitz cofLditiofL.Proof Let K 0 be &. given' real number,prime facloma.tioD :.ud 11PTe where PTeWe have:Z 2 be a. na.tural. Dumber which hu the ma.x {2, K} is a. prime n&.tuIILl number which does Dot divide :.18(:) - 8(Y)1 18 (p l pr; . ,pi;)- B (p 1 pr; . pi; .PTe) I is(pft) S(pf,) . S(pf,) - S(P!I) - S(pf,) - . - S(p!,) - S(pt)l·But : .PTe 11 which implies tha.t S(Pfj )18(:) - 8(Y)1 S(pfj)' for j 1, r so tha.t [S(P!I) - S(pf\)] [S(P!,) - S(Pf,)] . [S(pfJ - S(pfJ} S(pt) [S(Pft') - S(pf )1 [S(Pf/') - S(Pf,)] .1 [S(pf/'l) - S(pfJ] S(pt).

In )] it is proved the following formula. which gives a. lower a.nd a.n upper bound for5(l), wher p is a. prime na.tura.! number and . is a. na.tura.! number:r sing this formula.,we heLve:Pi·I-because p 2 .p; ' V)j L r .Then, we have:}8(;:;'1- 9(,;,ii:I\Ijt 5("1) (P1c -1)·;;·:;,' »":;",jI i(\ji(,J.I(-1).;:;. KI K(-:;r .::: ., :::'), K 'x i'11'Therefore we have proved that for every rea.! number K 0 there exist the natura.!numbers x, y such tha.t: .G( x) - d(y)1 K' x - y! which shows tha.t the function e doesnot verify the Lipschitz condition.Proposition 3 The /' 9 : .V· - .V, J(;;', S(pfl, :un.ere Pi ::re th.e prime1a.::':':''1.i 'Umoe:,s:unichliesmaiie":. -27l.;;::onci::J?i\ .:"!.Qtiiviie:::, ioes not verijy :he LipschitzfP"']Qf. Let K 0 be a glven rea.! number. Then for;; and 'j 2· ::: , using theTchebycheff theorem we know tha.t between: a.nd 'j there exists a prime natural numberp. It is clear ha.t p does not dividex and 2:::, thus ;(y) contains, in the sum, besides a.llthe terms of 9(;;), also S(p ) as a. term. We have:5(;:;) -6(;)1 !8(x) - 8(2:::)1 8(2:::) - G(:::) 5(::, - S(p:l) - G(x) S(p:l) {p - 1)y . 1 (p -1).2::: 1 x· 2;; -1 2;;2 -1therefore the function x· K K lx - yj9wo does not verify the Lipschitz condition.References)] Pal Gronas A praoj Qj ;he l.on-e.::.Hence ]f SAMMA , Smar&llda.che Function Journal,vol. 4-5, o.l, September 1994, pg.22-23.:2] C Popovici TeoMIl. nu.mereior , Editura. dida.ctid. i peda.gogica., Bucur ti, 1913.[3] F. Sma.randa.che An infinity of '.J.nsoivea p o biems concerning a. fu.nction in the nu.moertheory, Smara.nda.che Function Journal, voU, o.1, December 1994, pg.18-66. 4]T. Ya.n A problem oj ma.ximu.mSeptember 1994, pg.45.ISma.ra.nda.che Function Journal, vol.4-5, No.1,

PROPERTIES OF THE l'4'lJMERICAL FliNCTION Fsby I. Blilcenoiu, V. Seleacu, N. VirlanDepartament of Mathematics, University of CraiovaCraiova (1100), ROMANIAthis paper are studied some properties of the numerical functionFs(x):N - {O,I} 4' N Fs(x) L Sp(x), where 5 p(x) S(p%) is the SmarandacheInO pS.rp pnmefunction defined in [4].Numerical example: Fs(5) S(2 5 ) 5(35 ) S(55 ); Fs(6) S(26 ) 5(36 ) 5( ).It is known that: (p -1)r 1 5(pr) pr so(p -1)r S(pr) pr.Than(1)Where Jr( x) is the number of prime numbers smaller or equal with x.PROPOSITION 1: The sequence T( r) 1 log Fs (x) %1L --. has limit i 200.F:(l)Proof The inequality Fs(x) r(P2 ···4- P:n.r) - Jr(x» implies -logFs(x) -logr(A. P2 . · P1I(%) - n(r» -logr(n(r)A. - ;r(r» -logr -log 1Z'(x) -log(A. -1).Than for x i the inequality (1) become:1111-- ---FS(i) i(Pl "' P:(I) - Jr(i» i(PIJr(i) - Jr(i»iJr(i)(Pt -1).r1Than T(x) l-log(x)-logJr(x)-log(A-l) L - . - . - - . 2 m(I)(Pt -1).r1A 2 :- T(x) l-logx-logJr(x) L -.-.1 2 lJr(l) :- lim T(x);c "" 1 - lim logx-lim 10gJr(x) lim;c. "";c. ;c . ocPRoPOsmON 1. The equation Fs(x)6i: -. 1 . 1-oo-oo L -oo.1 2 I Jr( I) Fs(x 1) has no solution for x eN - to, I}.

Proof First we consider that x-I is a prime number with x 2. In the particular casex 2 we obtain Fs(2) 5(22) 4; Fs(3) 5(2 3 ) 5(3 3 ) 4 T 9 l3. So F2 (2) Fs(3).Next we shall write the inequalities:(2)U sing the reductio ad absurdum method we suppose that the equation Fs (x)has solution. From (2) results the inequalities Fs (x 1)From (3) results that: 7Z{x 1) 0.Ptr(x l) 7Z{x 1) so the diference from above is negative for x 0, and weobtained a contradiction. So Fs ( x) Fs ( x 1) has no solution for x 1 a prime number.Next, we demonstrate that the equation Fs ( x) Fs (x 1) has no solution for x andx 1 both composite numbers.ButLet p be a prime number satisfing conditions P x and P x - I. Such P exists2according to Bertrand's postulate for every x E - {O, I}. Than in the factorial of the numberp( x-I), the number p appears at least x times.So, we have S(pX) p(x -1).Butp(x-I) px p-x (if p x) and px p-x (p-I)(x I) I S(pX l).2Therefore :3 p x -1 so that S(pX) S(px l).Than Fs(x) S(pn ··· S(pX) ··· S(p X»Fs(x 1) S(ptl) . S(pX l) ··· S(p ;» Fs(x)In conclusion Fs ( x 1) Fs (x) for x and x 1 composite numbers. If x is a primenumber 7Z{ x) 7Z{ x 1) and the fact that the equation Fs ( x) Fs ( x 1) has no solution hasthe same demonstration as above.Finally the equation Fs (x) Fs (x 1) has no solution for any x EN - {O, I} .PROPOSITION 3. The function Fs(x) is strictly increasing function on its domain ofdefinition.The proof of this property is justified by the proposition 2.PROPOSITION 4. Fs(x y) Fs(x) Fs(Y) Vx,y EN - {O, I}.Proof Let X,Y EN - {O, I} and we suppose x y. According to the definition of FS(x)we have:7

F( X Y ) S( PIx y).T"'TS( P:r(x),rTY) . S( X"' Y).r"'V) T'P:r(.r) !'. S( P:ft )'S( P1f(y) 1.r Y)-r(4).r Y ) . S( P1f(x y)But from (1) we have the following inequalities:A (x Y)(Pt . P:r(.r) P:r(x) l . Pn(x y) -;r(x y» F(x y) andX(PI . P1f(x) - ;r(x» Y(PI . P:r(.r) "' P:r(X) "' Pn(y) - ;r(y» F(x) F(y) x(p,. ··· P1f(X» yep, . P,t(x) P1l(x) i . .' .» B(6)We proof that B A.B A ::: x(p, . P1l(r» Y(A . P1l(r» Y(PJZ(r) 1 . PJZ(y») x(A . P:r(.r» Y(A "' P1f(.r» x(P1l(r) l . P:r(r y» - xn(x y) Y(P1l(x) l . P1l(y» y( P.t(y) l . P1l(x y» - y;r(x y) ::: x(P1l(r) l "' P:r(x Y) - ;r(x y» Y(p,t(Y) i ···· . P1E{r y) - ;r(x y» But P:r(r y) ;r(x y) so that the inequality from above is true.CONSEQUENCE: FS(XY) Fs(x) Fs(Y)'if x,y eN -'{O,l}Because x andy eN - {O,l} and xy x Y than Fs(XY) Fs(x y) Fs(x) Fs(Y)PROPOsmON 5. We try to find lim FS(n)an- ooWe have Fs(n) LnS(pr') and:O p;SnP! pn1fll!A p,, "' P:(n) - ;r(n)PI P2 "' P1r(II)na - Ina - 1If a 1 thanlim n 1- a (A "' p1l(II)-;r(n»n- OO lim11- ""8(Pt "' p1f(II)-;r(n» co lim Fs( ) 00n

We consider now a 1.!!(n)L P, - ;r(n)We try to find lim LL P,and lim appling Stolz - Cesaro:a I--"I ::.JI a- Inn- oc!!( n)Let an;'!(n)P, - ;r(n)andbl1n- n a - 11 1:,(n l)Ln!!(n)PI-;r(n l)- L PI ;r(n)1 11 1 (n l)a-l n a - Iif (n 1) is a prime0, otherwise!!(n)Let cn L n a - I .Pi and d n1 1ThanCn 1 -L cn1 1(ndn 1 -dnn 1!!(n)PI-LPI!!(n l) 1 1 l)a-I n a - IP!!(n I)(n l)a-I - n a - I if(n l)a-1 n a - I(n I) is a prime0, otherwiseFirst we consider the limit of the function.limr- cox·I1m (x It-I - x a - I1HCO (a-l)[ (x l)a-2 - x a - 2 ]We used the l'Hospital theorem:In the same way we havelimr- cox 1(x l)a-I-x a- I 0So, for a 3 we have:. PI P2 "' P!!(n)-;r(n)hma-I 0r1m PI P2 a-I. P!!(n)Sonr- cor- conFinally lim F (:)r- ""n {- 0-0 00.andlim F(n)nar- cofor a 3for a 19 o.for a 3. 0Color a - 2 1

BmuOGRAPHYII) M. Andrei, C. Dumitrescu., V. Seleacu,L. TU\escu, t ZanfirSome remarks on the SmarandacheFunction, Smarandache Function Journal.Vol. 4, NO.1 (1994) 1-5;[2} P. GronasA note on S(p), Smarandache FunctionJournal. V. 2-3, No.1 (1993) 33 (3) M. Andrei. 1. BaIacenoiu. C Dumitrescu., A linear combination with SmarandacheE.lUdescu. N. Radescu. V. SeleacuFunction to obtain the Identity,Proceedings of 26m Annual IranianMathematic Conference Sbahid BabararUniversity of Kerman Kerman - IranMarch 28 - 31 1995[4) F. SmarandacheA Function in the Number Theory, An.Univ. Tuni Ser.StMat. VotXVIII,fase. 1(1980) 9, 79-88.

ON A LIMIT OF A SEQUENCE OF THENUMERICAL FUNCTIONby Vasile Seleacu, Narcisa VarIanDepartament of Mathematics, University of CraiovaCraiova (1100), ROMANIAIn this paper is studied the limit of the following sequence:n n1T(n) I-log O"s(n) I I.Ki IIe IO"S (Pi )We shall demonstrate that lim T (n) -x;.We shal consider define the sequence PI 2,P2 3, . ,Pn the nth prime number andthe function O"s:N*---)0N, O"s(x) I5(d), where 5 is Smarandache Function.d'xd OFor example: O"s(18) 5(1) 5(2) 5(3) 5( 6) 5(9) 5(18) 0 2 3 3 6 6 20We consider the natural numberp;,where Pm is a prime number. It is known that(p-l)r I5,5(pr)5,pr so 5(p'»(p-l)r.Next,wecanwriteIek(k l)O"S(Pi) (Pi -I), Vi212-----;-- - - - - O"s(p;) (Pi -l)k(k I)E{1, . ,m}, 'IlkE{I, . ,n}.This involves that:O"s(k»O, Vk?2 andp!5,p: ifa5,m and b5,n and p p: ifa c and b d.But O"s(P:) (Pm-l)n(n2 1) implies that -logO"s(p:) -IOg(Pm-l)n(n2 1)because log x is strictly increasing from 2 toNext, using inequality (I) we obtain X). L. 1 Ie 1 - Iog ( Pm - l)n(n l)T(Pmn) - 1- IogO"s (n)Pm L.2 l lle ! O"S(Pi )11

But Ie":l 2p",p", 1 2 2p",k(k l) p", 1i1e 1T(P!-) 1 log2-210gp",-log(p",-I) 1P1c - 1T(P!·) I Iog2 2(-IOgPm hi",1p-1b\Pt - I1c 1k.!.) 2P"'i I 2 .!.-log(p",-I)kPm 1 hi P1c - 1 t 1 kWe have r --:5: r -.P",l) 2rP.l(P,-"'--1 )-log(p",-l)(So: T(JJ!·) I log2-2 -logPm r 1e 1kPm 1Jc lkP. 1[ (P.L -k1)-Pm1] 1And then lim T(rm·):5:1 iog2 2lim(-logp", L -)- lim 2.--.",.,t 1 k",.hiP. 1 . [ 2 (P.L -1)] -.-lim log(p", -1) 1 log 2 2 hm (-log Pm r -)- lim - m- « P. « 1c 1 kP. p", 1t-Ik- lim log(p", -1) 1 log 2 2y-0-x -:c.P. ItISknown. (2 P.l)thatlim - - ' L - 0.p", 1 Jc 1 kIn conclusion lim T(n)P. ac00lim (-IOgPm P.-.aoIhi.!.) yk(Eulersconstant)and -co.REFERENCESTheory, An.Univ. Timi Ser.St.Mat. Vol.XVIII.fasc. 1(1980) 9, 79-88.[I) F. SmarandacheA Function in the Number[2)Smarandache Function Journal, NumberTheory Publishing Co, Phoenix,New-Yo Lyon, Vol. I, No.1, 1990.[3) PalGronasA proof the non-erislenceof "Somma",Smarandache Function Journal, V. 4-5,No.1 September (1994).

ON SOME SERIES INVOLVINGSMARANDACHE FUNCTIONbyEmil BurtonThe study of infinite series involving Smarandache function isone of the most interesting aspects of analysis.In this brief article me give only a bare introduction to it.EFirst we prove that the seriesK 2S(k)(kH) !converges and hasOE]e- 22' r2 lthe sumS (m) is the Smarandache function:Letusn L.Jic 21:'1!2!denoteS(k)(k 1)! 1n!1 - - . -12S(m) minby k 2nkimplies thatshowthatas follows:n WeEn'kE--,------;--,--(k 1)!S(k){kEN;.mlk!}Ek 2S(k)(k 1) !13 E . .(k l)!.,. . k,. .---,-nk 2 11(n l) !2!112(k l) ! 12

On the otherconsequently:handS (k)r. (k l)! ic ZS(k) (k :)!5E:: 1 -2 It follows thatL1nimpliesk 2- 3!4! - - n 12 S(k) 1 ,S(k) 2S{k}We can also check that anda -S (k)and52E .,- . thereforeEf e- 22LI1-2are strictly andkOE]e- 1 [ .Hence. S(k)In l'is a convergent series with sum(k l) !REMARK: Some of inequalitiesk .S(k)(k l) !Ek-2thatrENe(k-I) !S(k}(k r) !andrEN,are both convergent as follows:tk rS(k)(k-r)! ZlL.:. r(-2- . (n-r)O! 1!nWegetLie-rr 2r-: --,-(n --:-I}- - - - - . O!l!2!(n- I) !(k-r)!k rI 1Ik. (1 2- - I!! )2!. S{k)n-r )(n-I) ! IEn r En - r - 1which(k-r) !converges.S(k)Also we have(k r) !Let us define the setIfSo,m -Lm 300n! :xlS{m)!k 2IEN . {mEN: m-mSCm}! ! neN, n 3} .Itest n!2n!and thereforelDEHzA problem:LIit is obvious thatmEM.zSCm) n, theLk 2kENkS(k}! 00convergence behaviour of1S(k) !kEN,.theseries

R FERENCES1. Smarandache Function Journal, Vol.1 1990, Vol. 2-3, 1993,Vol.4-5 1994, Number Theory Publishing, Co., R. Muller Editor,Phoenix, New York, Lyon.Curent Address: Dept. of Math. University of Craiova,Craiova (1100)- Romania

OF TnE TYPE !SOME ?ROP ERTI ES OF SMARANDACHE FUNC T! ONSbyofasand [2]SI ype heDepar men ofUniversi yofSPC peiN!N- lN--pr i pS (Ie)ofwhichm)for .pape r his P,t.he seque nce(1. Propo s! tion.S1"\ ) 1"1 1"1iZPrPzfunc ionsSmar andac heindefin edarethe (1)s udy{max1. j rS p (i .Ie)}JJr hemono onicityofpi"' oper t.i essomeprese nt.edareshalland alsofunc tioni herefiefunc tions .,S (Ie)1""1.1.t.heseCralo vafollo ws:nInma hema icscons ruc ion hecons iderWeand Selea cu Vasi leIonaalaceno umono t.onic lt.yofeachof some subse quen ces ofIN-'The func tion Sn is mono tonou s incre asing for everyposi tiv integ er n.Proof .TheIe IeZ1Le Sfunc ionwhere1e,Ie1Z,isabvio uslymono t.onou se tN -, Supp osing that n is a prime numb erlcand t.akin gaccon t.t.ha incre asing .CS(1e z ))!n16 mult. iplenlc1- mult iple nZ

It. result.s t.hat. S (k )!S S (k ),1n 1 1 1cl" )Zm!S Sit. result.s t.hat.S11P.m k )SZso(k)Zmm.k )1(i .kP t)tZ(i.k)ttZn(Sp )i -monot.onousisfor every prime number p.t.wo nombersProof. For anyweP(iPS i s monot.onous i ncr easi ng.2. Proposition. The sequence of funct.ionsincreasing,SZJP jl"IS1J(i .k )} S(iis monot.onous increa-n(i . k )} JPm!S S(k)l"IS{{1 r.k)(iPmt.herefore SZi1IN,i eZi-2. iand forZn -anyhaveSP( n)i Sp(i. n)2. S Ci .pZn) SHence t.he sequence {Sp .p1}i -1t.herefore(n)S.1PZ!SS.1P ztis monot.onous increasing for everypr i me number p.3. Proposition.Let p and q t.woS (k)PProof. given prime numbers.S (k)qkEIN t.hen-Let. t.he sequence of coefficient.s (see (2]) aEvery k e INIf p q p p 2.Z,a,.,acan be uniquely writ.t.en as t. a (P 8172. p),.8(1)

whereThe0S tS p-l,for.of passingp ocedu et.C I.) ifist. can1from.-zt.owit.h ain formulek 1(1)t.t. is increaslng wit.h a unit.y and t.The procedure is cont.inued in t.hesame.-1is 0are not. i ncreasi ng wi t.h.-1isun t.y.wit.h a unit.y andSt.hen t.kS p,slncrease wit.h a unit.y,t.hennot.nei t.he t. , nor ::s t.1,s-1 ,andincreasingincreaslng( ,, ) i a uni t. yt.8-1 0way w. obt.ain t.heexpresion of k l. S p Ck l)Denot.ingwhen we pass fromdescribed above.Iet.o- SIe 1pt.he leap of t.he funct.ion S(Ie)correspondingIt. is abviously seenin t.h. case(,,)in t.h. cas.(;, ).A Sp)in t.h. casee i. i.i. AIcCSp)t.hat.:A Sp) Sen)PAnalogously we writ.eSen)qprocedureprocedure of passing from k t.o k lwit.h zero valueofSpweq0:a 0 SCi)P I'P p E,\Cn)lc 1Taking int.o account. t.hat. S (1) Pleapst.h.W. find t.hat.-'oft.opsSq(1)S C1) and using t.hededuceqt.hat.t.heis t.hen t.h.numbern ofleaps wit.h zero value of S respect.ively t.he number of' l.aps wit.hqval ue p of S i s 1 ess t.hen t.he numberp18of leaps ofSqwi t.h val ue

q resul ha E lc 1an heleaps z CJc)2S Ck)3 n e LN.give a tableandSafor 0 n 219 10 11 12 13 14 19 16 17 18 19 2020220022469\.PiCk)nz8893022000202228 10 12 12 14 16 16 16 16 18 18 20 22 243303330033303339 12 19 18 18 21 24 27 27 27 30 33 36 36 39 42 49S Ck)z S Ck)aI.S( ha Pi P z . p landPl1 lresul s .Pz\.Pz Sk 1, 2, . . . ,20.forincreasing sequence of prime numbersmono onous SS7For any . i h6i n(2)qq9HenceIf SCi)CS )464. Remark.k 1lc33303 24the leappqweE SCi) ppexample1kCS )S Cn) S Cn)HenceAslc S P i.Ck) l}1.0:)PjSPtthenCik)5. Proposition. If P and q are prime numbers and p.i q then S i. S .PqProof.Because p.iqi resultsSandFromS . C k)P(3) SCi k)Ppassing fromSip. C 1)kp. i q . S (1)q(3)S Ck)k toPk l, wededuce(4)i.)PTaking into account the proposition 3.hen .61 pass from tok l"'6119from (4)ob ainitresults that

(Spi) i (Sp) i. P'"iEandq41lc (S p. (SqJ(5)1e:1k:1Because.,E )havenn S P

Authors of papers concerning any of Smarandache type functions are encouraged to submit manuscripts to the Editors: Prof. C. Dumitrescu & Prof. V. Seleacu Department of Mathematics University of Craiova, Romania. The submitted manuscripts may be in the format of remarks, conjectures, solve

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Anatomy of a journal 1. Introduction This short activity will walk you through the different elements which form a Journal. Learning outcomes By the end of the activity you will be able to: Understand what an academic journal is Identify a journal article inside a journal Understand what a peer reviewed journal is 2. What is a journal? Firstly, let's look at a description of a .

excess returns over the risk-free rate of each portfolio, and the excess returns of the long- . Journal of Financial Economics, Journal of Financial Markets Journal of Financial Economics. Journal of Financial Economics. Journal of Financial Economics Journal of Financial Economics Journal of Financial Economics Journal of Financial Economics .

Andreas Wimmer, Indra de Soysa, Christian Wagner Number 61 Political Science Tools for Assessing Feasibility and Sustainability of Reforms ZEF – Discussion Papers on Development Policy Bonn, February 2003. The CENTER FORDEVELOPMENT RESEARCH (ZEF) was established in 1997 as an international, interdisciplinary research institute at the University of Bonn. Research and teaching at ZEF aims to .